Normalized defining polynomial
\( x^{36} - 136x^{27} - 1187x^{18} - 2676888x^{9} + 387420489 \)
Invariants
| Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(48526755753740305052512669329205843844959387036328330042655969\)
\(\medspace = 3^{90}\cdot 11^{18}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(51.70\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $3^{5/2}11^{1/2}\approx 51.70106381884226$ | ||
| Ramified primes: |
\(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(297=3^{3}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{297}(1,·)$, $\chi_{297}(131,·)$, $\chi_{297}(133,·)$, $\chi_{297}(263,·)$, $\chi_{297}(265,·)$, $\chi_{297}(10,·)$, $\chi_{297}(142,·)$, $\chi_{297}(274,·)$, $\chi_{297}(23,·)$, $\chi_{297}(155,·)$, $\chi_{297}(287,·)$, $\chi_{297}(32,·)$, $\chi_{297}(34,·)$, $\chi_{297}(164,·)$, $\chi_{297}(166,·)$, $\chi_{297}(296,·)$, $\chi_{297}(43,·)$, $\chi_{297}(175,·)$, $\chi_{297}(56,·)$, $\chi_{297}(188,·)$, $\chi_{297}(65,·)$, $\chi_{297}(67,·)$, $\chi_{297}(197,·)$, $\chi_{297}(199,·)$, $\chi_{297}(76,·)$, $\chi_{297}(208,·)$, $\chi_{297}(89,·)$, $\chi_{297}(221,·)$, $\chi_{297}(98,·)$, $\chi_{297}(100,·)$, $\chi_{297}(230,·)$, $\chi_{297}(232,·)$, $\chi_{297}(109,·)$, $\chi_{297}(241,·)$, $\chi_{297}(122,·)$, $\chi_{297}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{74}a^{18}-\frac{31}{74}a^{9}-\frac{1}{74}$, $\frac{1}{222}a^{19}-\frac{31}{222}a^{10}+\frac{73}{222}a$, $\frac{1}{666}a^{20}-\frac{253}{666}a^{11}+\frac{73}{666}a^{2}$, $\frac{1}{1998}a^{21}-\frac{919}{1998}a^{12}-\frac{593}{1998}a^{3}$, $\frac{1}{5994}a^{22}+\frac{1079}{5994}a^{13}+\frac{1405}{5994}a^{4}$, $\frac{1}{17982}a^{23}+\frac{1079}{17982}a^{14}-\frac{4589}{17982}a^{5}$, $\frac{1}{53946}a^{24}-\frac{16903}{53946}a^{15}+\frac{13393}{53946}a^{6}$, $\frac{1}{161838}a^{25}+\frac{37043}{161838}a^{16}-\frac{40553}{161838}a^{7}$, $\frac{1}{485514}a^{26}-\frac{124795}{485514}a^{17}-\frac{40553}{485514}a^{8}$, $\frac{1}{1728915354}a^{27}-\frac{796}{728271}a^{18}+\frac{49207}{728271}a^{9}-\frac{8445}{87838}$, $\frac{1}{5186746062}a^{28}-\frac{796}{2184813}a^{19}-\frac{679064}{2184813}a^{10}-\frac{96283}{263514}a$, $\frac{1}{15560238186}a^{29}-\frac{796}{6554439}a^{20}-\frac{679064}{6554439}a^{11}+\frac{167231}{790542}a^{2}$, $\frac{1}{46680714558}a^{30}-\frac{796}{19663317}a^{21}-\frac{679064}{19663317}a^{12}-\frac{623311}{2371626}a^{3}$, $\frac{1}{140042143674}a^{31}-\frac{796}{58989951}a^{22}+\frac{18984253}{58989951}a^{13}+\frac{1748315}{7114878}a^{4}$, $\frac{1}{420126431022}a^{32}-\frac{796}{176969853}a^{23}+\frac{18984253}{176969853}a^{14}-\frac{5366563}{21344634}a^{5}$, $\frac{1}{1260379293066}a^{33}-\frac{796}{530909559}a^{24}+\frac{18984253}{530909559}a^{15}-\frac{26711197}{64033902}a^{6}$, $\frac{1}{3781137879198}a^{34}-\frac{796}{1592728677}a^{25}+\frac{18984253}{1592728677}a^{16}-\frac{90745099}{192101706}a^{7}$, $\frac{1}{11343413637594}a^{35}-\frac{796}{4778186031}a^{26}+\frac{18984253}{4778186031}a^{17}+\frac{101356607}{576305118}a^{8}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{2108}{2593373031} a^{28} - \frac{31}{4369626} a^{19} - \frac{15467}{4369626} a^{10} + \frac{203391}{87838} a \)
(order $54$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $
Galois group
$C_2\times C_{18}$ (as 36T2):
| An abelian group of order 36 |
| The 36 conjugacy class representatives for $C_2\times C_{18}$ |
| Character table for $C_2\times C_{18}$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18^{2}$ | R | $18^{2}$ | $18^{2}$ | R | $18^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/31.9.0.1}{9} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{12}$ | $18^{2}$ | $18^{2}$ | $18^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{18}$ | $18^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $18$ | $1$ | $45$ | |||
| Deg $18$ | $18$ | $1$ | $45$ | ||||
|
\(11\)
| Deg $36$ | $2$ | $18$ | $18$ |